Coherent exciton transport in semiconductors

0.67 MB·English

by  Massimo Rontani

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0 Coherent exciton transport in
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l Massimo Rontani
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h CNR-NANOResearch Center on nanoStructures and bioSystems at Surfaces (S3),
- Via Campi 213a, 41125 Modena, Italy
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e L. J. Sham
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. Department of Physics, University of California San Diego,
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a Gilman Drive 9500, La Jolla, CA 92093-0319
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Acknowledgements
We thank Leonid Butov for critically reading the manuscript. This work is supported
by EU-FP7 Marie Curie Initial Training Network “Indirect Excitons: Fundamental
Physics and Applications (INDEX)”.

Contents
1 Coherent exciton transport in semiconductors 1
1.1 Introduction 1
1.2 Physical systems 5
1.3 Two-band versus BCS model 10
1.4 Andreev reflection at the interface between excitonic insulator
and semimetal 18
1.5 A perfect insulator 25
1.6 Josephson oscillations between exciton condensates in electro-
static traps 30
1.7 Conclusions 37
References 38
1
Coherent exciton transport in
semiconductors
1.1 Introduction
Anexcitonisaparticle-likeneutralexcitationofsolidsandmoleculescomposedofone
electronandoneholeboundtogetherbythemutualelectricalattraction[134,206,221,
228,286]. Its creationthroughinternalchargeseparationis mostfrequently causedby
the absorption of light and its demise is occasioned by electron-hole recombination,
mostly with emission of light and less frequently non-radiatively. The many-electron
ground state of the system, being an insulator, is immune to excitation until the
excitation energy reaches a threshold G known as the energy gap. When external
influences such as the electromagnetic field and lattice vibrations are ignored, the
exciton may be viewed as a robust state of an excited electron plus the hole which
has been left behind in the valence electronstates [230]. The hole acquiresits positive
charge from the loss of an electronic charge from the ground state whose total charge
is neutralized by that of the ions in the molecule or solid.
The photon-exciton interaction is responsible for the optical excitation (though
not necessarily in the visible frequency range) of the exciton and for its spontaneous
recombinationemittingaphoton(aquantumunitoflight).Thedipolematrixelement
responsible for the transition between the energy states is strong when the electron
andhole wavefunctions overlapin space or matchin wavevector.FromPlanck’slaw,
the frequency of the emitting light E /h is proportional to the energy loss E in
X X
returning the exciton state back to the ground state, with h being Planck’s constant.
If the constituent electron and hole of the exciton are mostly localized at an ion,
the exciton is localized, but with some probability to hop from site to site. Such a
Frenkelexcitonis commoninmolecules andmolecularsolids.At the other extreme,if
theelectronandholewavefunctionsarewidespreadasextendedorbitalsinamolecule
or Bloch waves in a crystal, their bound state as the exciton can have their center of
massmovingthroughthesystemwithease.SuchWannier(orWannier-Mott)excitons
are most common in broad-band and small-gap semiconductors (a semiconductor is
distinguished from an insulator qualitatively by a smaller energy gap, with the fre-
quency of the emitting light from the exciton spanning the range from visible light to
very far infrared). Wannier excitons resemble the hydrogenatom or,more closely,the
positronium system composed of an electron and a positron. Because of the dielectric
screening of the electrical force in small-gap materials and sometimes the small effec-
tive mass of the electron, the Wannier exciton radius is 10 to 100 times larger than
the positronium radius, which is approximately 0.1 nm.
Coherent exciton transport in semiconductors
2
Excitons,beingmadeoftwofermions,behaveasbosonsonthescalelargerthanthe
exciton radius and therefore may macroscopically occupy a single quantum state [20,
175,40,125,3,124,77].Iftheexcitonlifetimeislongenoughtoallowforreachingquasi-
equilibrium,thediluteandcoldgasofopticallygeneratedexcitonsmayundergoBose-
Einstein condensation (BEC) [50,87,198]. The critical temperature for exciton BEC,
oftheorderof1Kfortypicaldensitiesinsemiconductors,isbasicallythetemperature
atwhichthethermalDeBrogliewavelengthbecomescomparabletotheaverageinter-
exciton separation. The possibility of achieving BEC of excitons by shining light on
solids hasbeen thoroughlyinvestigatedin the lastfifty years(see the reviews[99,178,
87,157,176,143,237,30,31,32,112,240]). Semiconductors are particularly appealing
for this goalas they may provide excitons with a a lifetime (hundreds of ns in bilayer
structures [30]) longer than the time required for cooling.
In an indirect-gap semiconductor such as silicon, where the momentum of the
exciton does not match that of the photon, the excitons are generally formed after
relaxation of optical excitations with initial energy much higher than the gap. The
indirect exciton has a long lifetime because its recombination with the emission of a
photon requires the conservation of momentum to be satisfied by the assistance of a
latticevibrationortrappingbyadefect.Consequently,the excitonshavetimetoform
a largepoolknownasanelectron-holedrop(see the reviews[209,122]).Alternatively,
thedelayinopticalrecombinationmaybeduetothesymmetryofthecrystal,asinthe
direct-gap oxide Cu O which has conduction and valence bands of like parity hence
2
the opticaldipolar transitionis forbidden [110]. The chapter by Kuwata-Gonokamiin
Volume 1 focuses on the aspects of Bose-Einstein condensation of optically generated
excitons in semiconductors.
In a direct-gapmaterial,the spatial separationof the electron andhole can be en-
forcedbyhousingthemintwolayerssufficientlyclosetomaintaintheirelectricattrac-
tion[232,160,71]. The recombinationofsuchindirectexcitons maythenbe controlled
by changing the electronand hole wave function overlapwith an electric field [5,300].
Aninterestingphenomenonisthelaserspotexcitationoftheseindirectexcitons,lead-
ing to the formationof two concentric luminous circles centered at the laser spot plus
other localized bright spots randomly placed between the circles. Whereas the forma-
tion of the inner ring [34] is due to the migration of indirect excitons away from the
laser spot as optically inactive excitons, the localized bright spots [34] as well as the
outer ring[34,238,37,204]formonthe boundariesbetweenelectron-richandhole-rich
regions.Atlowtemperature the outerringisanecklaceofevenlyspacedbrightspots,
whoseoriginisnotfullyunderstood[103].Thissystemofexcitonsinadoublequantum
well is considered a good candidate for condensation. These experiments and related
work in double quantum wells are reviewed in [237,30,31,32,256,112].
Anotherpossible—andelusive—mechanismofcondensationofexcitonsasbosonsis
that excitons form spontaneously at thermodynamic equilibrium even in the absence
of an optical excitation. Such process signals the transition to a permanent phase
known as excitonic insulator (EI) [177,134,123,55,114], which is originated by the
instabilityofthenormalgroundstateofeitherasemiconductororasemimetalagainst
the spontaneous formation of bound electron-hole pairs. The wave function of the
strongly correlated EI ground state is formally similar to that proposed by Bardeen,
Introduction
3
Cooper, and Schrieffer for superconductors [16]. As a matter of fact, both excitons
and Cooper pairs are absent except as fluctuations in the normal high-temperature
phase and form only in the ordered, low-temperature phase—respectively the EI and
the superconductor. Besides, both condensation of excitons and that of Cooper pairs
arebestdescribedinthereciprocalspaceofthecrystalsolid.TheEIphaseisreviewed
in [96,135,99,50,186,87,157,176,237,156,112,240].
It is intriguing to observe that condensation of other types of bosons composed of
two fermions leads to spectacular manifestations of quantum mechanical coherence,
such as the superfluidity ensuing from the pairing of 3He atoms [149], Fermi alkali
atoms confined in optical traps [21,80], nucleons in neutron stars [79,196], the super-
conductivity induced by Cooper pairs in metals [54], and the non classical momenta
of inertia in nuclei [169,23]. The above phenomena may regarded as distinct realiza-
tions of superfluidity, associated to the coherent, dissipationless flow of charge and /
or mass. However,excitons are neutral and stay dark unless recombine radiatively,as
shown in Table 1.1, which compares the distinct features of the condensates made of
composite bosons. The signature of the macroscopic order of the exciton condensate
is, at present, controversial for the superfluid transport but its other manifestations
will be discussed next.
The aim of this Chapter is to illustrate some recent theoretical proposals con-
cerning the detection of coherent exciton flow [212,213,214]. The reader may refer to
the literature reviewed in Sec. 1.2 for a discussion of the conceptual and experimen-
tal difficulties inherent in the realization of exciton condensates. Here we set aside
such difficulties and adopt in a pedagogical way the simplest mean-field description
of the condensate,on which we lay our theoreticaldevelopmentin order to detect the
transport properties of the exciton condensate.
In particular, we focus on the exciton analogues of two phenomena, i.e., Andreev
reflectionandJosephsoneffect,whicharehallmarksofsuperconductingbehavior,and
stressthecrucialdifferencesbetweenexcitonsandCooperpairs.Ourfirstmainconclu-
sionis that the excitonic insulator is the perfect insulator in terms of both chargeand
heat transport,with an unusually high resistance at the interface with a semimetal—
the normal phase of the condensed state. Such behavior, which should be contrasted
with the highelectricalconductance ofthe junction betweensuperconductor andnor-
malmetal, may be explainedin terms ofthe coherenceinduced into the semimetalby
theproximityoftheexcitoncondensate.Thenweshowthattheexcitonsuperflowmay
be directly probed in the case that excitons are optically pumped in a double-layer
semiconductor heterostructure: we propose a correlated photon counting experiment
for coupled electrostatic exciton traps which is a variation of Young’s double-slit ex-
periment.
We last mention that, due to the interactionbetween electrons and light, not only
cananexcitondecayirreversibilyintoaphotonorviceversa,butitcanalsoexchange
roles with the photon in a quantum-mechanically coherent fashion. Thus, the exciton
mayexistinthe solidinthe superpositionstate ofanexcitonandaphoton,knownas
polariton.Whereasthe photonenergyvarieslinearlywithits momentum atthe speed
of lightin the vacuum, the exciton energydepends on the squareof its center-of-mass
momentum.Forsmallmomenta,theexcitonandthephotoncanapproximatelymatch
Coherent exciton transport in semiconductors
4
Table 1.1 Excitonic insulator (EI) versus superconductor `a la Bardeen-Cooper-Schrieffer (BCS).
The interface referred to in the Table is the junction between normal and condensed phase. For a
general discussion of the condensates made of composite bosons see [136]. For specific EI features
see [114] (Meissner effect), [297] (superconductivity), [298] (superthermal conductivity), [212,213]
(Andreevreflection), and [214] (Josephson oscillations).
Physical property Excitonic insulator BCS-like superconductor
Nature of the composite boson Exciton Cooper pair
Boson charge Neutral 2e
Boson momentum Crystal momentum Crystal momentum
(commonly ignored in the free
electron gas approximation)
Boson mass Effective mass Effective mass of the electron
quasiparticle in the Fermi level
region (of thickness provided by
phonon Debye frequency)
Type of long-range order Diagonal Off-diagonal
Superfluidity ? Superconductivity
Meissner effect No Yes
Superthermal conductivity No No
Nature of the quasiparticle Electron (hole) Bogoliubon
Andreev reflection Yes Yes
Interface electric conductance Decreased Increased
Interface thermal conductance Decreased Decreased
Proximity effect Yes Yes
Josephson oscillations Yes Yes
both their momentum and energy values, the coupling mixing the two states into two
superpositions of photon and exciton with an energy splitting. Thus, the massless
photon is slowed down by the massive exciton by virtue of the quantum-mechanical
superposition.TheChapterbyYamamotoinVolume1dealswithaspectsofpolariton
condensation.
The structure of this Chapter is the following: After a review of previous work
(Sec. 1.2), in Sec. 1.3 we illustrate the mean-field theory of the EI emphasizing its
relationwiththe BCStheoryofsuperconductors.We thenintroduce the phenomenon
of Andreev reflection in Sec. 1.4 and analyze its observable consequences in Sec. 1.5.
Section 1.6 on the Josephson effect ends the Chapter.
Physical systems
5
1.2 Physical systems
This sectionbrieflyreviewsrecenttheoreticalandexperimentalworksonexcitoncon-
densation, focusing on diverse physical systems. Without attempting an exhaustive
review, we refer the reader to more comprehensive essays whenever available.
1.2.1 Bose-Einstein condensation of optically generated excitons
The pursuit of Bose-Einstein condensation of optically generated excitons in semi-
conductors, which dates back to the sixties, presently focuses on both classic systems
such as Cu O and novel low dimensional structures (for reviews see [99,178,87,157,
2
176,143,237,120,30,31,156,32,256,112,240,141,224]). A very active field concerns
“indirect”excitons.Suchexcitonsaremadeofspatiallyseparatedelectronsandholes,
hosted in two quantum wells that are sufficiently close to maintain electrical attrac-
tion between the carriers of opposite charge. This setup has several advantages: (i)
The overlap of electron and hole wave functions is controlled by applying an elec-
tric field along the growth direction of the bilayer heterostructure, thus increasing
the exciton recombination time by orders of magnitude with respect to the single-
well value [5,300]. (ii) The confinement effect along the growth direction increases
the exciton-phonon scattering rate, improving exciton thermalization [292]. (iii) The
dipolar repulsion among indirect excitons disfavors the formation of biexcitons and
electron-holedroplets[38,156,246,222,267,148,269,47]aswellaseffectivelyscreensthe
in-planedisorderpotential[111,220,102,107,208,7].(iv)Astheelectricfieldparallelto
thegrowthdirectionmaybelaterallyvariedusingsuitablylocatedelectrodes,onemay
tailorthe in-planeeffectivepotentialsforexcitons,thus realizingartificiallycontrolled
traps[109,97,44,86,75,102,107,223,104,7,8],ramps[94,74],lattices[295,296,208,207],
“exciton circuits” [101,106,88], and “exciton conveyers”[279].
Exciton traps may also be created by means of the uncontrolled in-plane disor-
der of the double quantum well [299,36,34,102,107], the strain experienced by the
heterostructure [257,119,183,179,268,285], the laser-induced confinement [98,9], the
magnetic field [46]. The realization and control of exciton traps is a key capability to
reach exciton BEC: As the long range order in two dimensions is smeared by quan-
tum fluctuations, a weaker requirement for the macroscopic occupation of the lowest
exciton level is that the exciton coherence length exceeds the trap size [31].
The present evidence of exciton BEC is based on distinct features of the emit-
ted light (photoluminescence, PL) that appear at low temperature: (i) The PL dy-
namics exhibits bosonic stimulation of the scattering of hot optically dark excitons
into optically active low-energy states [35]. (ii) The PL signal becomes noisy in a
broad range of frequencies, as it occurs in the presence of coherence [38,139,142].
(iii) The degree of polarization of the emitted light increases with decreasing tem-
perature [144,145,103], consistently with gauge symmetry breaking. (iv) The exciton
mobility is enhanced, which may be attributed to superfluid behavior [33]. (v) The
radiativedecayrate increases,whichmay be explainedinterms of“superradiance”of
amacroscopicdipole[33]orcollectivebehaviorattheonsetofcondensation[144,145].
(vi) The PL lineshape narrows and departs from the Maxwell-Boltzmann distribu-
tion [139,144,145,146], as it may be expected for the macroscopic population of a
single exciton state.